A192017 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the Fibonacci tree of order n (1<=k<=n; entries in row n are the coefficients of the corresponding Wiener polynomial).

1, 2, 1, 4, 4, 2, 7, 10, 9, 2, 12, 21, 27, 15, 3, 20, 40, 65, 57, 25, 3, 33, 72, 138, 163, 114, 37, 4, 54, 125, 270, 394, 378, 206, 54, 4, 88, 212, 500, 854, 1033, 796, 354, 74, 5, 143, 354, 891, 1716, 2479, 2463, 1571, 574, 100, 5, 232, 585, 1545, 3265, 5424, 6559, 5469, 2917, 896, 130, 6

Offset: 1

Author

Emeric Deutsch, Jun 21 2011

Keywords

• nonn
• tabl

Comments

The Fibonacci trees f(k) of order k are defined as follows: 1. f(-1) and f(0) each consist of a single node. 2. For k>=1, to the root of f(k-1), taken as the root of f(k), we attach with a rightmost edge the tree f(k-2). See the Iyer & Reddy references. These trees are not the same as the Fibonacci trees in A180566.

Sum of entries in row n is A191797(n+2).

Examples

			T(2,2)=1 because in the Fibonacci tree of order 2, namely /\, there is only 1 pair of nodes at distance 2 (the two leaves).
Triangle starts:
   1;
   2,  1;
   4,  4,  2;
   7, 10,  9,  2;
  12, 21, 27, 15,  3;
  20, 40, 65, 57, 25,  3;
		

References

• K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of Binomial trees and Fibonacci trees, Int'l. J. Math. Engin. with Comp., Accepted for publication, Sept. 2009.

Links

• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
• K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of binomial trees and Fibonacci trees, arXiv:0910.4432 [cs.DM], 2009.

Crossrefs

Cf. A000071, A011973, A165910, A191797.

Programs

Maple

G := (1+t*z)/(1-z-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do r[n] := sort(coeff(Gser, z, n)) end do; w[0] := 0; w[1] := t; for n from 2 to 11 do w[n] := sort(expand(w[n-1]+w[n-2]+t*r[n-1]*r[n-2])) end do: for n from 1 to 11 do seq(coeff(w[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form

Mathematica

m = 11; Gser = Series[(1 + t*z)/(1 - z - t*z^2), {z, 0, m}]; Do[r[n] = Coefficient[Gser, z, n], {n, 0, m}]; w[0] = 0; w[1] = t; Do[w[n] = Expand[w[n - 1] + w[n - 2] + t*r[n - 1]*r[n - 2]] , {n, 2, m}]; Flatten[Table[Coefficient[w[n], t, j], {n, 1, m}, {j, 1, n}]] (* Jean-François Alcover, Sep 02 2011, after Maple *)

Formula

T(n,1) = A000071(n-2) (Fibonacci numbers minus 1).

Sum_{k=1..n} k*T(n,k) = A165910(n) (the Wiener indices).

The Wiener polynomial w(n,t) of the Fibonacci tree of order n satisfies the recurrence relation w(n,t) = w(n-1,t) + w(n-2,t) + t*r(n-1,t)*r(n-2,t), w(0,t)=w(1,t)=0, where r(n,t) is the generating polynomial of the nodes of the Fibonacci tree of order n with respect to the level of the nodes (for example, r(2,t) = 1 + 2t for the tree /\; see A011973 and the Maple program).